In addition to the contrast invariance of orientation selectivity, there are several other properties of simple cells that must be addressed by any model of cortical function. First, cortical responses saturate as the contrast of a stimulus increases. The responses do not, however, saturate at a fixed firing rate determined by the electrical properties of the cell, but at a rate that changes with the stimulus (Dean, 1981, Albrecht and Hamilton, 1982, Maffei et al., 1973). Responses to stimuli of non-optimal orientation or spatial frequency saturate at lower firing rates than do the responses to optimal stimuli. Second, as the contrast of a stimulus increases, the time course of the responses of simple cells changes. Specifically, the temporal phase (or latency) of the response to a sinusoidal grating advances in time (Dean and Tolhurst, 1986, Albrecht, 1995, Carandini et al., 1997). Third, responses to gratings of high temporal frequency increase more with increasing contrast than do responses to gratings of low temporal frequency. As a consequence, the temporal frequency tuning of simple cells changes with contrast (Albrecht, 1995). Finally, the response to a superposition of two stimuli is often less than the sum of the responses to each stimulus alone, even when one of the component stimuli evokes no response at all. A prominent example of this effect is cross-orientation inhibition: Two gratings, one of the preferred-orientation and one of the orthogonal orientation, evoke a smaller response than does the preferred-orientation grating alone (Morrone et al., 1982, Hammond and MacKay, 1981, Bauman and Bonds, 1991, Ferster, 1981, Bishop et al., 1973, Bonds, 1989, De Valois et al., 1985, Geisler and Albrecht, 1992, Nelson, 1991, Gulyas et al., 1987, Li and Creutzfeldt, 1984, DeAngelis et al., 1992)
The normalization models were proposed (Albrecht and Geisler, 1991, Heeger, 1992) to explain some of these effects. In these models, the feedforward geniculate input is assumed to grow linearly with stimulus contrast, but is divided or normalized just prior to threshold by an inhibitory input whose strength also increases with contrast. The combination of the inhibition and excitation yields a sigmoidal, saturating function of contrast. Carandini and Heeger (1994) proposed that the normalization signal would take the form of a shunting inhibition driven by the pooled responses of surrounding neurons of many different preferred orientations and spatial frequencies. This shunting inhibition, which would thus be orientation independent but increase with stimulus contrast, would increase the conductance of a cell. This shunt would lower the cell's membrane time constant, thereby lowering its integration time, with the resulting effect of advancing the phase of responses to sinusoidal gratings and enhancing responses to higher temporal frequencies (Carandini and Heeger, 1994, Carandini et al., 1997). Finally, a shunting inhibition at all orientations could explain cross-orientation inhibition.
While the normalization models can be made to fit the spike responses of simple cells, two key predictions have not been borne out in intracellular experiments. First, the models require a large stimulus-evoked shunting conductance that depends only on stimulus contrast, and hence is independent of stimulus orientation. It now seems clear, despite initial indications to the contrary (Ferster and Jagadeesh, 1992, Douglas et al., 1988), that visual stimuli do indeed evoke large increases in the input conductance of simple cells (Anderson et al., 1999, Borg-Graham et al., 1998, Hirsch et al., 1998). The amplitudes of stimulus-induced conductance changes are strongly dependent on orientation, however, with a preferred orientation and tuning width similar to that of the membrane potential responses (Anderson et al., 1999). Second, the membrane time constant does not change sufficiently to explain contrast-dependent changes in temporal properties. Normal resting time constants are in the range of 20 ms (Anderson et al., 1999, Hirsch et al., 1998). The normalization model, however, requires decreases in time constant of 60 ms or more, which is clearly impossible since time constants cannot go below zero.
How, then, can we explain the response properties that are described by the normalization framework? It turns out these properties can be explained in the context of the feedforward model with push-pull inhibition by a variety of existing nonlinearities in the electrical properties of cells and the temporal properties of synaptic inputs (Kayser et al., 1999). These include saturation and contrast dependent phase advance in the responses of the LGN inputs themselves, frequency-dependent synaptic depression, spike-rate adaptation, non-zero threshold, and small stimulus-induced changes in membrane time constant. Since LGN inputs saturate with contrast (Cheng et al., 1995, Sclar, 1987), we need only explain the difference between LGN and cortical saturating contrasts. This can be accounted for by synaptic depression (Markram and Tsodyks, 1996, Stratford et al., 1996), which causes the geniculate synaptic input to saturate before presynaptic LGN firing rates saturate (Abbott et al., 1997, Tsodyks and Markram, 1997). Synaptic depression, together with the other nonlinearities listed above, can also explain the difference between LGN and cortical phase advances (Chance et al., 1998) and changes in temporal frequency tuning with contrast. Synaptic depression and spike-rate adaptation act as contrast-dependent high-pass filters, allowing high-temporal-frequency responses to grow with contrast at a faster rate than low-temporal-frequency responses. Contrast-induced decrease in the membrane time constant also causes a relative enhancement of high-temporal-frequency responses with increasing contrast. Finally, cross-orientation inhibition and other two-stimulus suppression effects can be accounted for, in part, by the push-pull inhibition in the feedforward model: the non-preferred grating evokes strong push-pull inhibition, reducing responses to the preferred orientation (Krukowski et al., 1998).
It should be noted that even if the feedforward model with push-pull inhibition and same-phase-excitation were to prove largely correct, it is still at best incomplete. A number of issues need to be addressed. The model does not currently deal with direction selectivity, for example, though it is likely that the addition of lagged and non-lagged input using the scheme proposed by Saul and Humphrey (1992) would fit well with the current model. The model also does not deal with the diversity of cortical inhibitory interneurons. Studies in rat somatomotor thalamocortical slices show two classes of inhibitory neurons in layer 4 (Gibson et al., 1999): a feedforward class receiving strong thalamic input, like the inhibitory neurons of the model, and a feedback class receiving little or no thalamic input, which is not incorporated in the model. Furthermore, cells of each class show strong gap junction coupling among themselves. Intracellular studies in cat V1 layer 4 have found some inhibitory neurons that respond primarily to stimulus contrast: they are complex cells (not selective for stimulus polarity) and unselective or only weakly selective for stimulus orientation (J. Hirsch, private communication). Their role is unknown. While the simplicity of the model gives order to a wide variety of findings, the complexity of the cortical circuit should not be underestimated.